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Follow-up (2026-09-27): more stale "open" labels, found by syncing with sources that maintain status data. Again, this is only meant to save others duplicated effort.

1. Erdős problems. All 632 EP-* entries are labelled open in v1.6.0. As of today, erdosproblems.com lists 90 of them as resolved (we fetched each page; many resolutions are recent, several by AI systems, and most are Lean-verified). Grouped by the site's status:

• Proved, Lean-verified (28): EP-38, EP-123, EP-126, EP-152, EP-258, EP-281, EP-283, EP-330, EP-351, EP-358, EP-369, EP-457, EP-469, EP-557, EP-571, EP-610, EP-750, EP-793, EP-825, EP-865, EP-987, EP-997, EP-1014, EP-1022, EP-1051, EP-1071, EP-1096, EP-1129
• Proved (9): EP-380, EP-591, EP-652, EP-851, EP-863, EP-986, EP-1021, EP-1105, EP-1130
• Disproved, Lean-verified (14): EP-1, EP-43, EP-74, EP-90, EP-125, EP-146, EP-180, EP-193, EP-533, EP-846, EP-847, EP-884, EP-990, EP-1092
• Disproved (8): EP-92, EP-543, EP-574, EP-575, EP-705, EP-869, EP-960, EP-992
• Solved otherwise (e.g. determined/answered), Lean-verified (12): EP-42, EP-119, EP-183, EP-190, EP-202, EP-318, EP-619, EP-650, EP-694, EP-696, EP-741, EP-888
• Solved otherwise (e.g. determined/answered) (19): EP-320, EP-321, EP-346, EP-387, EP-421, EP-477, EP-603, EP-625, EP-633, EP-690, EP-730, EP-783, EP-858, EP-896, EP-920, EP-948, EP-1005, EP-1089, EP-1091

The site's machine-readable status file (teorth/erdosproblems, data/problems.yaml) may be the easiest way to keep these entries in sync.

2. Ben Green's 100 open problems. The current version of the list (updated December 2025) marks these as solved. Note that the dataset's GREEN-xxx numbers differ from the numbering in Green's list.

3. OWR-1452-012 (Zhao's Vanishing Conjecture for homogeneous quartics). Zhao proved that this conjecture, over all n, is equivalent to the Jacobian conjecture over all n (Trans. AMS 359 (2007), arXiv:math/0409534). The July 2026 Jacobian counterexample (see SMA-016 above) therefore refutes it for some n; we have not identified the smallest such n.

— Alper Ferudun

Follow-up 2 (2026-09-27): Kourovka Notebook, issue 21 (KOU-21.*). The arXiv version of the notebook updated today (arXiv:1401.0300v46) marks the following issue-21 problems as solved (asterisk), while v1.6.0 still labels them open:

(The other nine starred issue-21 problems, 21.8, 21.12, 21.14, 21.15, 21.18, 21.24, 21.43, 21.58, 21.150, are already marked solved in the dataset.)

— Alper Ferudun

Follow-up 3 (2026-09-27): KOU-21.68 — new counterexample (our own result, unrefereed). Kourovka Notebook Problem 21.68 (M. Kida) conjectures that every finite semi-abelian group is monomial. This is false. There is a semi-abelian group of order 768 = 2^8·3 with a non-monomial irreducible character of degree 8:

• Ĝ = B ⋊ W, where W = E ⋊ A₄ is the index-two subgroup of C₂ ≀ A₄, T ≅ SL(2,3) ≤ W is the binary tetrahedral group acting on the eight quaternion units, and B is the augmentation (even-weight) submodule of the permutation module F₂[W/T].
• General reduction (Clifford theory): if a linear character of the abelian normal subgroup N has stabiliser T in W and T has a non-monomial irreducible character, then N ⋊ W is not an M-group.
• Checked by exact computation, including every subgroup of index 8 and the full character table of Ĝ (exactly the three degree-8 irreducible characters are non-monomial). Scripts are in the source archive.

Preprint: https://doi.org/10.5281/zenodo.23000305 · paper page: https://eulersolve.org/papers/kou-21-68/

Suggested label: solved (answered negatively). It has not been peer-reviewed yet, so independent checks are welcome. Minimality of the order is not claimed; Kida's Magma search covered all orders up to 240.

— Alper Ferudun

Follow-up 4 (2026-09-27): internal status inconsistency. For 188 problem numbers, the literature-triage block inside the record itself ("Literature review (checked 2026-08-17)", Status: solved, Classification: SOLVED-IN-LITERATURE) disagrees with the status field, which still says open (same in v1.6.0 and v1.7.0). 94 of them are already covered in the comments above; the remaining 94 are listed below by source so they can be reconciled in one pass. We have not re-verified each triage conclusion ourselves, and some of them refute a literal wording rather than the intended question, so they are pointers, not claims.

• HIL (1): HIL-009
• SMA (1): SMA-004
• NT (5): NT-008, NT-023, NT-035, NT-037, NT-086
• GREEN (3): GREEN-061, GREEN-064, GREEN-100
• GEO (3): GEO-004, GEO-006, GEO-028
• GEOM (1): GEOM-026
• GT (1): GT-008
• GRAPH (4): GRAPH-006, GRAPH-008, GRAPH-046, GRAPH-049
• TOP (1): TOP-003
• ALG (2): ALG-033, ALG-036
• COMB (2): COMB-012, COMB-004
• HL (1): HL-F
• GUY (3): GUY-A8a, GUY-A11, GUY-A15
• KP (6): KP-1.51, KP-3.14, KP-4.37, KP-4.125, KP-5.9, KP-5.15
• OPG (51): OPG-23298, OPG-50149, OPG-37185, OPG-426, OPG-1797, OPG-37167, OPG-37181, OPG-37230, OPG-692, OPG-37086, OPG-610, OPG-37341, OPG-34908, OPG-37081, OPG-37089, OPG-37218, OPG-37316, OPG-37364, OPG-57613, OPG-59952, OPG-59997, OPG-824, OPG-46606, OPG-47285, OPG-59911, OPG-2242, OPG-59994, OPG-616, OPG-36939, OPG-52200, OPG-47031, OPG-47643, OPG-37305, OPG-47646, OPG-677, OPG-690, OPG-691, OPG-735, OPG-177, OPG-157, OPG-732, OPG-760, OPG-2379, OPG-37444, OPG-37863, OPG-37402, OPG-655, OPG-1783, OPG-37245, OPG-37295, OPG-57401
• EP (9): EP-129, EP-520, EP-524, EP-545, EP-550, EP-612, EP-638, EP-654, EP-996 — note that erdosproblems.com still lists all nine as open (EP-550 as "open (Lean)"), so these triage conclusions deserve a second look before relabelling.

A JSON list (problem number, status field, triage status) is available on request.

— Alper Ferudun

Hello, thank you for these detailed additions! They are now integrated.

Follow-up 5 (2026-09-28): two new results, one problem already solved in the literature, and corrections to Oberwolfach and Kourovka records (stale statuses, transcription errors, merged records, one misattached note). Each item was checked against the current version (commit 2ea030b). Where a page is given, it was also checked in the report itself; literature was checked against arXiv and Crossref. Suggested changes are in bold.

New results (unrefereed preprints with full proofs, verification scripts and independent referee checks)

• OWR-12861-021 → solved (no). Heinig's Question 2 (OWR 01/2014, p. 82) has a negative answer for every odd n ≥ 7: K_{(n+1)/2,(n−1)/2} with a perfect matching (or a matching plus one P₃) inside the larger side has minimum degree ⌈n/2⌉ but no spanning copy of the near-square, under both readings of "periphery"; at n = 9, K_{4,4,1} is a counterexample under every reading. For n = 7 the host is Heinig's own graph X (arXiv:1112.5101, Def. 28). The cycle-space Question 1 is not affected (Hou–Yin). Paper: doi:10.5281/zenodo.23004012 (page).
• KOU-21.76 → solved. The existence question was first answered by Ya. N. Nuzhin, Sib. Math. J. 67 (2026) 840–845, Corollary 1 (examples in every characteristic, over F(y,z)); the notebook (v46) does not record this yet. Our note gives explicit examples with short proofs, including one-variable examples over F₃(t) (which also satisfy the hypotheses of 19.48), and shows that in characteristic 3 such nets exist over K iff K is not algebraic over F₃. Paper: doi:10.5281/zenodo.23004034 (page).

Stale statuses

• OWR-12330-013 → solved (yes). King answers his own Question 1 on the next page: "The answer to this is yes" (OWR 02/2013, p. 105). K_k with k pendant vertices at every vertex has χ_f = k > 3k/4 + 1 for k ≥ 5.
• OWR-1319-022 → solved (no). Two word metrics on H₃(ℤ)×ℤ have ratio → 1 but unbounded difference: Breuillard, Groups Geom. Dyn. (2014) (arXiv:0704.0095), and Breuillard–Le Donne, PNAS 110 (2013), §5, who cite this report.
• OWR-5158-016 → solved (yes). By Deroin–Hurtado (arXiv:2008.10687, Thm 1.3), irreducible lattices in finite-centre semisimple groups of real rank ≥ 2 are not left-orderable, so any cocompact arithmetic lattice in SL(3,ℝ) works. See also Witte Morris's exposition (2026).
• OWR-14213-006 → solved. The statement is the unit-interval (Hessenberg) form of the Stanley–Stembridge conjecture, proved by Hikita, J. Amer. Math. Soc. (2026) (arXiv:2410.12758).
• OWR-1386-016 → solved, if "threshold" has its usual coarse meaning (the constant b is then immaterial). This is the two-graph Kohayakawa–Kreuter conjecture: the 1-statement is due to Mousset–Nenadov–Samotij (CPC 2020), and Christoph–Martinsson–Steiner–Wigderson (Proc. LMS 130, 2025) completed the proof. A sharp threshold at b·n^(−1/m₂) would be a different question; we could not re-read the wording in OWR 48/2006.
• OWR-14299911-005 → solved (disproved). The source itself reports Chalopin–Chepoi's counterexample to the "MSO decidable ⇔ grid-free" conjecture (ACM TOCL 20 (2019)). Please also drop the second sentence: the special-cube-complex result concerns Thiagarajan's other conjecture, and the counterexample itself comes from a virtually special complex.
• OWR-17294-009 → at least partially solved. Part (1), Byott's question, is answered no by Di Matteo–Ferrara–Trombetti, arXiv:2607.22795 (July 2026 preprint). The record bundles Vendramin's Problems 1, 3 and 5 (OWR 51/2019, p. 3222); splitting it would help.
• KOU-21.115 → solved (yes). Sambale, arXiv:2609.09052 (September 2026 preprint), proves |G∖U| ≥ |G|/2ⁿ whenever n left cosets have union U ≠ G, and right cosets are left cosets of conjugates. The notebook (v46) has not starred it yet.
• OWR-4213-002 → solved. The statement is the Feit–Thompson theorem (Pacific J. Math. 13 (1963)). Also, "nontrivial" simple groups should read "non-abelian".
• OWR-1265-024 → not an open problem. The source poses it as a puzzle and gives the answer C(a+b,a) − C(a+b,a−1), which counts standard (b,a)-tableaux (G. James, OWR 15/2006, pp. 965–966).
• Pointer only, OWR-11568-005. Lutowski's Theorem 2 (Publ. Math. Debrecen 99 (2021)) says that the holonomy of a non-torus Kähler flat manifold has at least two distinct irreducible constituents. Since h^{1,1}(A/G) = dim End_G(T₀A), this seems to force b₂ ≥ 2 for free torus quotients of dimension ≥ 2. Worth a check.

Transcription errors

• OWR-12330-010, -011, -012: the ceilings in the source are missing (OWR 02/2013, p. 103, Conj. 3; p. 104, Conj. 4–6). As written, C₅ violates all of them: the right-hand sides are 5/2 or 3 − ε, while χ(C₅) = 3.
• OWR-1189-007: Kohl's Conjecture 2 reads ⌊3d(1−1/n)⌋ + 1, not ⌈·⌉ (OWR 7/2006, p. 416). As written it is false for d = 1 and n ≥ 4, because χ^{1,1}_ℓ(P_n) = ch(P_n²) = 3.
• OWR-5154-009: the source puts the profinite closure on the product [g₁]^F⋯[g_k]^F, not on N(g₁,…,g_k) (OWR 26/2011, p. 1452, Conj. 2). As written, the answer is trivially no.
• OWR-12007-008: with "r, s ∈ ℝ" (the same wording as OWR 35/2012, p. 2173) the question is false for w = a, since R(a,r,s) = r. The source's own results are for rational r, s, so this should read r, s ∈ ℚ.

Merged records

• OWR-4791-001 merges two conjectures that the organisers' introduction (OWR 1/2011, p. 6) reports as proved: Simonovits–Sós (Ellis–Filmus–Friedgut) and Sumner (for large n, Kühn–Mycroft–Osthus). Splitting it would help.
• OWR-9790352-039, OWR-10252930-031, OWR-9790352-036: the statement is already clean, but original_statement still contains a neighbouring item, respectively:
  • the Plummer–Zha conjecture, proved during the workshop (OWR 1/2022, p. 71; Chudnovsky–Seymour, JGT 103 (2023));
  • a weighted Turán conjecture, proved by Bradač (arXiv:2205.08923);
  • Narayanan's permanent inequality (OWR 1/2022, pp. 69–70), which is open and has no record of its own.

Misattached note

• OWR-12861-020: the literature note (Hou–Yin, arXiv:2503.15950) is about Heinig's Question 1 (OWR 01/2014, p. 81), which has no record of its own. It does not concern the Diestel or Friedgut questions in this record, and the record's "partially solved" label rests only on that note.

Correction to Follow-up 2. arXiv:1401.0300v46 (Kourovka Notebook No. 21) was posted on 1 September 2026, not on 27 September as I wrote there; the list of starred problems is unaffected.

— Alper Ferudun

Follow-up 6 (2026-09-28): three new results, and corrections to Oberwolfach combinatorics records (stale statuses, questions answered in their own source, transcription errors, duplicates, merged records, garbled extracts and misattached notes). Each item was checked against the current version (commit 2ea030b). Where a page is given, it was also checked in the report itself; literature was checked against arXiv and Crossref. Suggested changes are in bold.

New results (unrefereed preprints with full proofs, verification scripts and independent referee checks)

• OWR-16164-019 → solved (no). Sportiello's conjecture B_λ ≥ A_λ (OWR 23/2018, pp. 1455–1457) fails for the band λ = {(x,y) ∈ [7]² : −3 ≤ y−x ≤ 4}, where A_λ = 536 and B_λ = 515. Every digitally convex shape of side ≤ 6 satisfies the inequality; among the 5,693,968 shapes of side 7, only this band and its transpose violate it. (The record attaches the red and blue crosses to the opposite colours; that swap is a bijection on colourings and does not change B_λ.) Paper: doi:10.5281/zenodo.23006715 (page).

• OWR-1782-009 and OWR-1386-013 (the same conjecture) → false as stated. Both ask that minimum codegree ⌊(n−k+3)/2⌋ force a tight Hamiltonian cycle for all n: for n ≥ k+1 ≥ 4 in OWR 01/2008 (p. 46), and with no range of n in OWR 48/2006 (p. 2928). This all-n form is Conjecture 1.1 of Rödl–Ruciński–Szemerédi, Adv. Math. 227 (2011), who attribute it to Katona–Kierstead. It fails for (k,n) = (3,7), (3,9), (4,8), (5,9) and (5,10). The smallest counterexample is an apex joined to all 15 pairs of a 6-set W, plus the ten faces of the hemi-icosahedron on W. Its pair-degrees are 3 and 5. But in a cyclic order (apex, w₁, …, w₆) the disjoint windows w₁w₂w₃ and w₄w₅w₆ would both have to be faces, and no two faces are disjoint. The (5,10) example has codegree 4 = (n−k+3)/2, so the version without the floor fails too. The large-n statement is not affected: RRS proved it for k = 3, and Letzter–Lang–Ranganathan–Sanhueza-Matamala have announced it for all k ≥ 3, as reported in arXiv:2609.08613. Suggested status: "false as stated (small counterexamples); the large-n form is proved for k = 3 and announced for all k". Also:

  • OWR-1386-013 should say "tight" Hamiltonian cycle, as its source defines it; under a Berge reading these hypergraphs do have Hamiltonian cycles;
  • mark the two records as duplicates;
  • the literature DOIs on OWR-1782-009 concern other thresholds (10.1016/j.jcta.2015.01.004: Hamilton ℓ-cycles with ℓ < k/2; 10.1112/jlms.12561: minimum d-degree conditions). RRS 2011 and arXiv:2609.08613 are the relevant references.

  Paper: doi:10.5281/zenodo.23006718 (page).

• OWR-14299577-018 → solved (yes, both questions). The problem was proposed by C. Bernert and N. Arala Santos, in the problem session compiled by T. F. Bloom (OWR 51/2025, p. 2756), so proposed_by can be filled in. Let A ⊂ {−n,…,n}∖{0} contain exactly one of k and −k for every k ≤ n. Then at most two elements of [1,n] are missing from A − A. This is sharp: {1,…,a} ∪ {−(a+1),…,−n} misses exactly a and a+1 whenever n/2 ≤ a ≤ n−1. For every B ⊆ [1,n], the number of pairs (a,b) ∈ A² with a − b ∈ B is at least ⌊(|B|−1)²/4⌋, and this is attained for every |B| ≤ ⌊2n/3⌋+1. So for |B| ≥ εn the count is at least (1/4 − o(1))ε²n². The key step is that the representation counts r(d) of any set S of differences of one parity satisfy Σ_{d∈S} r(d) ≥ C(|S|,2), by a double count of the positions where the sign pattern repeats at distance d. The record's original_statement is garbled: it drops the set-up sentence and splices in two sentences from the preceding problem (Assing). Paper: doi:10.5281/zenodo.23006720 (page).

Stale statuses

• OWR-16160-016 → solved as stated (classical). Valette asks whether the congruence subgroups of G ⊂ SL_N(ℤ) depend on the embedding (OWR 19/2018, p. 1151). In general they do. For example, F₂ embeds as ⟨(1 2; 0 1), (1 0; 2 1)⟩ and as ⟨(1 3; 0 1), (1 0; 3 1)⟩. The level-2 subgroup of the second embedding contains no congruence subgroup of the first, and the level-3 subgroup of the first contains none of the second, so the two congruence topologies are incomparable. Classical instances go back to Serre: SLₙ(ℤ) has congruence subgroups whose images under the adjoint map are not congruence subgroups (Lubotzky–Venkataramana, Algebra Number Theory 13 (2019), Prop. 2.1). For the groups of the talk, Z² ⋊_A Z, the answer is the opposite, and the same holds for every solvable G. Solvable groups have the congruence subgroup property (Chahal, Nagoya Math. J. 79 (1980)), and solvable subgroups of GLₙ(ℤ) are polycyclic, so every embedding induces the profinite topology (LV, §1.1). A scope note would help.
• OWR-1265-006 → solved (yes). Olsson's containment question (OWR 15/2006, p. 913) is the Olsson–Stanton conjecture. Vandehey proved it (arXiv:0809.2134, 2008), and Fayers gave another proof (JCTA 118 (2011)).
• OWR-1536-011 → solved (no, both parts). Knox (arXiv:1212.3345) gives a hypergraph on which Breaker, moving second, wins Maker–Breaker, yet Chooser wins Chooser–Picker. Knox notes that the Picker–Chooser part is equivalent to it; the equivalence comes from the transversal-hypergraph duality stated in the source (OWR 20/2007, p. 1095). So the transversal hypergraph of his example refutes the Picker–Chooser part too.
• OWR-4425-020 (duplicate -021) → solved (yes). Shallit's Conjecture 48 (OWR 37/2010, p. 2236) is Theorem 18 of Cassaigne–Currie–Schaeffer–Shallit, J. ACM 61 (2014), for exactly this morphism.
• OWR-2090-021 → partially solved. Adiprasito–Björner (arXiv:1401.7301, Thm 2.1) prove the Mikhalkin–Ziegler conjecture from this problem session:
  • for a generic weight and t ≤ min{0, total weight}, the proper flats of weight > t form a homotopy Cohen–Macaulay poset, hence an (r−3)-connected one;
  • the "non-negative" version follows by a small perturbation;
  • they credit rank 3 to Pinchasi–Ziegler (personal communication, 2008);
  • shellability is still open (their Open Problem 3.2).
• OWR-12481-012 → solved (no). When Friedgut re-posed it for fixed t and large n, the report recorded: "This turns out to be false; a counterexample was found by Gábor Tardos" (OWR 22/2016, p. 1217). The question as posed here already fails for n = 4, t = 2. Pair σ with σ∘(1 2) if σ maps {1,2} onto {1,2} or {3,4}, and with σ∘(3 4) otherwise. The resulting 12 two-cosets refine none of the 8 partitions of S₄ into 1-cosets.
• OWR-16633-025 → solved (no), over every field. We found no written answer, but a 4-element example settles it: T₀ = F², T₁ = ⟨e₁⟩, T₂ = ⟨e₂⟩, T₃ = ⟨e₁+e₂⟩. Then f(0) = 2, f(1) = f(2) = f(3) = 1, and f(S) = 2 for every other nonempty S. Suppose f = Σ c_j r_j with c_j > 0 and matroid ranks r_j on {0,1,2,3}; the r_j need not even be representable.
  • Every matroid has r(0i) ≥ r(0) and r(ij) ≤ r(i) + r(j). Since f attains equality in both, so does every r_j.
  • So in each r_j, the elements 1, 2 and 3 lie in the closure of 0, which has rank ≤ 1, and at most one of them is a non-loop.
  • Hence r_j(123) = r_j(1) + r_j(2) + r_j(3) for every j, and summing gives f(123) = 3. But f(123) = 2.
  • Over GF(2), 2·r(U₂,₄) is another example. Dougherty–Freiling–Zeger (arXiv:0910.0284, §4) represent it over every field. Its only possible matroid summands are copies of U₂,₄, which is not binary.

Answered in the source itself

• OWR-4791-006 → solved. Fox–Lee–Sudakov prove both conjectures in the abstract itself (OWR 01/2011, pp. 11–12: Thm 2, f(m) = ⌊√(4m+1)⌋ − 1, and Thm 4). The paper is Israel J. Math. 191 (2012). The record's own verification note already says so.
• OWR-16931-001 → solved. The record asks only about sufficiently large n. For that case the source says "We answer this affirmatively for all sufficiently large n" (OWR 19/2019, p. 1158). This is Glock–Joos–Kim–Kühn–Osthus, JEMS 23 (2021).
• OWR-14299518-002 → solved. Alon answers both questions in the abstract itself (OWR 42/2025, pp. 2249–2250): (1) no (Thm 3), (2) yes (Thm 4). These are Erdős problems #664 (disproved) and #732 (proved). The note's "structural characterization" is not part of this record.
• OWR-12872-010 → solved, and one formula needs fixing. Stanley (with F. Liu) proves both of Elkies' conjectures in the same abstract (OWR 12/2014, pp. 696–697); the paper is Ramanujan J. 36 (2015). For n = 2m the count of maximum families is 2^((m−1)(m−2))·(2^m − 1), not ·(2m − 1). A brute-force count gives 28 for n = 6 and 960 for n = 8.

Transcription errors and literal readings

• OWR-14298158-004: Claesson's conjecture is monotonicity in the length n for fixed k: |Av_n^k(1324)| ≤ |Av_{n+1}^k(1324)| (OWR 6/2024, p. 284; Claesson–Jelínek–Steingrímsson, JCTA 119 (2012)). The same correction applies to the Av(1324, 231) part. As written (k → k+1), the statement is false for every n, e.g. |Av_{2,1}| = 1 > 0 = |Av_{2,2}|.
• OWR-17135-030, -031, -032: the source prints M_ii = Σ_{S∋i}|S| (OWR 39/2019, p. 2464). This is a typo for Σ_{S∋i} X_S = deg(i).
  • With that diagonal, det M = |X|·|Y|·τ(G)/∏_{y∈Y} deg y. So Conjecture 5 becomes equivalent to Ehrenborg's Conjecture 4, as the source says.
  • As printed, Conjecture 5 is false. Take X = [3], hang 13 leaves on each vertex of X, and add one vertex joined to all three. Then det M = 850 > 512 = det(diag M). It also fails if the sum runs only over the sets S that occur.
  • With the corrected diagonal, Conjecture 5 is Ehrenborg's conjecture, proved by Ho (arXiv:2603.17997; -030 already cites it).
  • The refinement in -031/-032 is false as literally stated, already at n = 3. There det(diag M) − det M is homogeneous of degree 3 with coefficient −1/4 on X₁₂X₁₃X₂₃. No such polynomial equals Σc_μx^μ + Σc_μν(x^μ − x^ν)² with all c ≥ 0, because the square terms would contribute a nonzero part of even degree. If multipliers x^ρ(x^μ − x^ν)² were intended, the record should say so.
• OWR-14299089-007: as written, the question is trivial: (a_n b_n)² ≥ a_{n−1}a_{n+1}b_{n−1}b_{n+1} for nonnegative sequences. The source asks Brändén–Ferroni–Jochemko's Question 6.1 (Trans. AMS 2026, arXiv:2408.12386). Write Σ p(n)xⁿ = W(p)/(1−x)^(deg p+1). If W(p) and W(q) are log-concave with no internal zeros, is W(pq)?
• OWR-11695867-010: as the source itself says (OWR 57/2022, p. 3274), the case ℓ = 0, Σ f_λ² = n!, is "exactly the Robinson-Schensted-Knuth algorithm". Louf's Open problem 1 is a bijective proof of n!·H_{n,ℓ} = Σ_{λ⊢n} f_λ² C_λ^ℓ for all ℓ. Here H_{n,ℓ} counts ℓ-tuples of transpositions with product 1, and C_λ is the content sum.
• OWR-16164-005: the literal question has a classical answer. Put y_i = 1 − x_i for odd i; each constraint then becomes y_i ≤ y_{i+1} or y_i ≥ y_{i+1}. So the polytope is the order polytope of a fence, a poset whose Hasse diagram is the path 1–2–⋯–n. Its Ehrhart polynomial is Ω(P, t+1) (Stanley, DCG 1 (1986)), so h* is the descent polynomial of its linear extensions (natural labelling). We checked this for all 63 sign sequences with n ≤ 6. Mark solved, or restate if another interpretation was intended.

Duplicates

• Across reports: OWR-12481-012 is the Friedgut half of OWR-12861-020 (OWR 18/2013, p. 1118; OWR 01/2014, p. 81). Splitting OWR-12861-020 would leave there only Diestel's k-block question, which is open.
• New: OWR-14299904-003 repeats -002, which is Weigandt's Conjecture 1 (OWR 2/2026, p. 133); -004 is Conjecture 2.
• Already flagged: the following records are marked as repeats in their own statement_verification, but are still published as open or partially solved. So each of these problems is counted twice. A duplicate status, or unpublishing, would help. The records are OWR-734-010, OWR-1183-007, -012, OWR-4425-019, -021, OWR-4791-016, OWR-4798-022, OWR-11136-011, OWR-14604-015, OWR-16633-022, -024, OWR-16763-026, -028, OWR-17135-032 and OWR-1703876-017.

Merged records

• OWR-1703876-016 (duplicate -017) bundles Problems 6–10 of the OWR 30/2020 problem session (p. 1522), posed by five different people. Splitting would help. Two of its notes need fixing:
  • Problem 10 (Welzl, partial triangulations) is solved (yes) by Kupavskii–Volostnov–Yarovikov, Europ. J. Combin. 108 (2023) (arXiv:2104.05855). -017 lists that arXiv paper under the names Aichholzer–Orden–Schnider.
  • Problem 8 (Steiner, bichromatic triangles) is still open. -016 says a 2026 preprint proves it. In fact Radtke–Keszegh–Lauff (arXiv:2601.20574) prove it only for at most 5 red pseudolines (so for n ≤ 11). In general they prove only that a two-coloured triangle or quadrangle exists.
• OWR-4425-003 bundles Currie's Open problems 2–4 (OWR 37/2010, p. 2206): Restivo–Salemi reachability, the curling-number conjecture and the lexicographically least 5/2-power-free word. Splitting would help.
• OWR-2090-006 bundles Linial's open-ended challenges on Latin squares and the Γ function with one precise conjecture (OWR 44/2008, p. 2496): the maximum rank of a real n×n×n tensor is (1+o(1))n²/2. That conjecture deserves its own record.
• OWR-2090-026: the statement is clean (Problem 9, Barvinok–Samorodnitsky), but original_statement also contains Problem 10 (Welzl, spanning trees versus triangulations). That problem is already OWR-2090-027.

Garbled extracts (all already marked "unrecoverable", but still published as open or partially solved)

• OWR-12861-024: the source itself is ill-posed (OWR 01/2014, p. 83). The matrix has rows indexed by S_n but columns only by {σ : LIS(σ) ≥ n−t}, and it is called both M and A, so its determinant is undefined.
• OWR-9790358-007: the opening of Hakopian's abstract plus its title (OWR 7/2022, p. 405), with no question in it. Its only conjecture, Gasca–Maeztu, is OWR-9790358-001.
• OWR-12697684-002, -003: table-of-contents lines (OWR 1/2023, pp. 9–10). -012: background sentences from Bucić's abstract (p. 42) about the Erdős–Hajnal conjecture, which is -011.
• Also in OWR 1/2023: the "original OWR report" link of OWR-12697684-001, -002, -003, -011 and -012 points to 10.4171/owr/2022/57 (Enumerative Combinatorics). It should be 10.4171/owr/2023/1.
• OWR-14298158-001: a table-of-contents entry (OWR 6/2024, p. 277). -016: the closing remark of an abstract on Bevan's conjecture (p. 299); the conjecture itself is -015.
• OWR-723-001: Beck's four circle-discrepancy questions. The source itself reports them as answered, by Schmidt and by Beck (OWR 13/2004, pp. 678–679). The conjecture that remains is OWR-723-002. Mark solved or remove.

Misattached notes

• OWR-2090-021: the note (parametric assignment, rotation matching) belongs to Rote's Problem 2 of the same session, least-squares matching under rotation (OWR 44/2008, pp. 2546–2547). The relevant literature for this record is Adiprasito–Björner (above).
• OWR-4425-012: the note ("sum-square avoidance", Au–Robertson–Shallit) is about the additive-square problem, Problem 47, which is OWR-4425-018. The record itself is Shallit's Problem 38 on pattern characterisations of α-powers (OWR 37/2010, p. 2231).

— Alper Ferudun

OPG-361: preprint evidence for both matroid-flat conjectures

This is an attributed literature update, not a new result or manuscript by us.

• Log-concavity: Matt Larson, Counterexamples to two conjectures about matroids, Example 2.2 and Theorem 2.3, gives a rank-76 graphic matroid with W_74^2 < W_73 W_75.
• Unimodality: Alexander Divoux, Chayim Lowen and Shouda Wang, Matroid flat counts are not unimodal, Theorem 1.2, gives counterexamples using q-lifts of Larson's family. This answers the other universal clause of OPG-361 negatively.

These statements concern flats / Whitney numbers of the second kind, not characteristic-polynomial coefficients or independent-set counts. The unimodality reference also updates the limited claim in follow-up 80 (the AMR-156-0026 paragraph) that Rota's unimodality conjecture was still open; no new audit of that whole multi-part record is claimed.

We read the relevant proofs and reproduced the flat-count arithmetic. As a concrete instance of the published construction, t=28 and q=59 gives a rank-83, 5,016-element matroid with W_79 > W_80 < W_81. This specialization is not claimed novel or minimal.

The inspected sources are arXiv preprints. Please retain the dataset's v1.8 unrefereed_claim distinction rather than treating this notice as evidence of peer review. Our check is AI-assisted and self-audited, not independent human review or formal verification.

Alper Ferudun · Mercury Software GmbH

Status follow-up 141: 2 status corrections from the Kourovka Notebook (Problems 12.9 and 17.68) and a note on one Oberwolfach record

• Each item was checked against the current dataset (commit d371ee9, release v1.8.0), against its source (the Kourovka Notebook, arXiv:1401.0300v48 of 6 October 2026 and v49 of 20 October 2026, cited by problem number; the Oberwolfach report) and against the cited papers as far as they are openly available; publication data are from Crossref. Suggested changes are in bold.

Partial results in print (labels → partially solved)

• KOU-12.9 → partially solved (part b answered yes in a refereed survey, whose proof rests on a preprint; part a holds for actions on ℝⁿ-trees); the dataset currently labels it open. Kourovka Notebook, Problem 12.9 (A. M. Gaglione, D. Spellman, 12th issue, 1992; v48, p. 60, unchanged in v49): call a group tree-free if it acts freely without inversion on a Λ-tree for some ordered abelian group Λ; a) must every finitely generated tree-free group satisfy the maximal condition for abelian subgroups? b) the same for finitely presented tree-free groups. A group satisfies the maximal condition for abelian subgroups exactly when all its abelian subgroups are finitely generated (the union of an ascending chain of abelian subgroups is abelian). For b): O. Kharlampovich, A. G. Myasnikov, D. Serbin, Actions, length functions, and non-Archimedean words, Internat. J. Algebra Comput. 23 (2013), no. 2, 325–455 (doi:10.1142/S0218196713400031), Theorem 62(1) in the numbering of arXiv:1211.3207v2: in a finitely presented Λ-free group every abelian subgroup is free abelian of finite rank, at most the rank of the abelianization. The survey says this answers Question 2, p. 250, of I. Chiswell's book Introduction to Λ-trees (2001) for finitely presented groups. The proof given there uses either the structure theorem for finitely presented groups with a regular free length function in Λ (Theorem 57: iterated HNN extensions of a free group along maximal abelian, finitely generated subgroups) together with the embedding Theorem 60, or Theorem 58 (every finitely presented Λ-free group is ℝⁿ-free) together with Guirardel's theorem below; Theorems 57, 58 and 60 are cited from the authors' preprint arXiv:0911.0209, listed in the survey as submitted, and we found no journal publication of it. For a) and Λ = ℝⁿ with the lexicographic order: V. Guirardel, Limit groups and groups acting freely on ℝⁿ-trees, Geom. Topol. 8 (2004), no. 3, 1427–1470 (doi:10.2140/gt.2004.8.1427), proves that a finitely generated group acting freely on an ℝⁿ-tree is built from free abelian and surface groups by finitely many free products and amalgamations over cyclic groups, and that its abelian subgroups are finitely generated (abstract). For Λ = ℤⁿ the question is easy: an abelian group acting freely without inversion on a Λ-tree embeds in Λ (it acts by translations along a common axis), so every abelian subgroup of a ℤⁿ-free group is finitely generated (standard; our remark). Neither v48 nor v49 comments on the problem. Open as far as we found: a) for general Λ. Read: the problem in v48 and v49; the overview of results on finitely presented Λ-free groups (pp. 10–11) and the statements and proofs of Theorems 57–63 (pp. 118–120) in the survey (arXiv:1211.3207v2); the abstracts of arXiv:0911.0209 and of Guirardel's paper.
• KOU-17.68 → partially solved (|G : C| is finite whenever G is residually finite, residually periodic, linear, hyperabelian (for example soluble) or a subgroup of a word-hyperbolic group); the dataset currently labels it open. Kourovka Notebook, Problem 17.68 (V. D. Mazurov, 17th issue, 2010; v48, p. 118, unchanged in v49): if C is a cyclic subgroup of G and G = CFC with F a finite cyclic subgroup, is |G : C| finite? As F is finite, G = ⋃_{f ∈ F} CfC is a union of finitely many double cosets of C, so the problem is a special case of M. J. Dunwoody's question whether a group that is a finite union of double cosets of a cyclic subgroup H must have |G : H| finite (Problem 5 in Combinatorial and Geometric Group Theory, London Math. Soc. Lecture Note Ser. 204, Cambridge Univ. Press, 1995, p. 322, as cited by Jabara 2006); if C is finite, G is finite (our remark). G. A. Niblo, Double coset decompositions of groups, J. Algebra 220 (1999), no. 2, 512–518 (doi:10.1006/jabr.1999.7935), shows that such a G is virtually cyclic if it is residually finite or word hyperbolic (zbMATH review Zbl 0944.20024); then |G : C| is finite. E. Jabara, Gruppi che sono unione di un numero finito di laterali doppi, Rend. Sem. Mat. Univ. Padova 116 (2006) 41–53 (open access on Numdam), proves that |G : H| is finite whenever G is a union of finitely many double cosets of H and G is residually finite with H virtually nilpotent (Teorema 1), G is residually periodic with H finitely generated and virtually nilpotent-by-polycyclic (Teorema 2), G is linear over a field with H finitely generated and soluble (Teorema 6), G is hyperabelian with H virtually polycyclic (Teorema 7), or G is a subgroup of a word-hyperbolic group with H virtually cyclic (Teorema 8); each applies with H = C. E. Jabara, Actions of abelian groups on groups, J. Group Theory 10 (2007), no. 2, 185–194 (doi:10.1515/JGT.2007.014), Theorem 2, as stated in the zbMATH review Zbl 1137.20022: if H is virtually cyclic and G is a union of finitely many double cosets of H, then |G : H| is finite, or G has a characteristic subgroup R such that G/R and Z(R) are finite and R/Z(R) is an infinite simple group; the review adds that it is not known whether the second case occurs. So in Problem 17.68, |G : C| is finite unless G has such a subgroup R. Neither v48 nor v49 refers to these papers. Open as far as we found: groups with such an infinite simple section. The works citing Niblo's or Jabara's 2007 paper that OpenAlex lists (A. Minasyan 2005 on hyperbolic groups, J. Howie, J. Algebra 2009, on soluble groups, Jabara, J. Algebra 2010, on abelian automorphism groups with finitely many orbits) do not, by their titles, abstracts or reviews, decide this case; we did not read them in full. Read: the problem in v48 and v49; Jabara 2006 (introduction and the statements of Teoremi 1, 2, 6, 7 and 8); the zbMATH reviews of Niblo 1999 and Jabara 2007 (neither paper was read: the J. Algebra site refused our automated access, and the J. Group Theory paper is not openly available).

Notes on the records (labels unchanged)

• OWR-1784-002 (assessment): U. Görtz (joint work with T. Haines, R. Kottwitz, D. Reuman), "Affine Deligne Lusztig varieties", Oberwolfach Report 3/2008, pp. 136–139, Conjecture 3 (p. 138; G split, semisimple and simply connected): a) for x in the shrunken Weyl chambers, X_x(1) ≠ ∅ if and only if η₂(x)⁻¹η₁(x)η₂(x) lies in no W_T with T ⊊ S, and then dim X_x(1) = ½(ℓ(x) + ℓ(η₂(x)⁻¹η₁(x)η₂(x))); b) for every σ-conjugacy class [b] there is n₀ with X_x(b) ≠ ∅ ⟺ X_x(1) ≠ ∅ whenever ℓ(x) ≥ n₀. The assessment says that the nonemptiness direction remains open; part a) has since been proved. It is the case b = 1 of Conjecture 9.5.1(a) of U. Görtz, T. J. Haines, R. E. Kottwitz, D. C. Reuman, Compos. Math. 146 (2010) 1339–1382 (arXiv:0805.0045), whose Proposition 9.5.4 gives the emptiness direction (Theorem 4 of the report). U. Görtz, X. He, Doc. Math. 15 (2010) 1009–1028 (arXiv:1006.2291), Theorem 3.1.2, prove nonemptiness for basic b and x in the shrunken Weyl chambers (the lowest two-sided cell, Section 2.3 there), and X. He, Ann. of Math. 179 (2014) 367–404 (arXiv:1201.4901), Corollary 12.2, proves the dimension formula for G simple and split (both papers work with adjoint groups, simple in the results used, and explain why the general case reduces to these). X. He, Forum Math. Pi 9 (2021) e9 (arXiv:2001.03325), Section 1.3, records 9.5.1(a) as "established" in the 2014 paper. Part b) is the nonemptiness half of Conjecture 9.5.1(b), where x need not be shrunken. For shrunken x it follows from Corollary 6.4 of the 2021 paper: for G simple, x in a shrunken Weyl chamber and dominant translation part λ_x ≥ ν_b + 2ρ^∨, X_x(b) ≠ ∅ if and only if κ(b) = κ(x) and η₂(x)⁻¹η₁(x)η₂(x) lies in no proper W_T, which for simply connected G is the criterion of a); for G simple, ℓ(x) ≥ n₀ forces the bound on λ_x (our remark). For x outside the shrunken Weyl chambers we found no result; Section 1.3 of the 2021 paper notes that later works add the shrunken assumption. As printed, b) needs G simple: for SL₂ × SL₂, b = (1, b₂) with b₂ not basic and x = (x₁, 1) with ℓ(x₁) odd, X_x(1) ≠ ∅ by the report's SL₂ example, while X_x(b) = X_{x₁}(1) × X_1(b₂) = ∅, since a point of X_1(b₂) would make b₂ σ-conjugate to an element of I, hence to 1 (our remark). The label partially solved stays; suggest updating the assessment. Read: the abstract in the report; the introduction and Section 9.5 of the Compositio paper; Sections 1, 2.3 and 3.1 of Görtz–He; Section 0.7, Corollary 10.4 and Section 12 of He (2014); Sections 1 and 6 of He (2021) in both versions.

The remarks marked as ours are short and given in full; they have not been refereed. Two of the papers used for KOU-17.68 are known to us only from their zbMATH reviews, and the result used for b) of KOU-12.9 rests, according to the survey that states it, on a preprint without journal publication; this is stated in each item. These are AI-assisted, self-audited checks, not independent human peer review: we read the source and the cited statements and took publication data from Crossref. The suggested labels are suggestions. Corrections are welcome. — Alper Ferudun, Mercury Software GmbH

AIM-PROBABILITY-0034: a newer critical mixing upper bound

This is an attributed literature update, not a new solution or manuscript by us. The v1.8 record's assessment still quotes the upper bound O_d(n^(3+4/(d-2)) log n).

Yan Ru Pei, Critical and near-critical influence bounds for ferromagnetic Ising models, Corollary 1.3, gives O(n^(2+2/(d-2)) log n) for zero-field random-scan heat-bath dynamics at beta_c = atanh(1/(d-1)), on every graph of maximum degree d >= 3. Set epsilon=0 in that corollary. This therefore applies to the random regular graphs in this record.

The order n^(3/2) lower bound in random-scan steps follows from Prodromidis and Sly, Corollary 1.3, whose continuous-time convention updates each vertex at rate one. The upper and lower bounds do not match; retain partially solved rather than claiming a sharp critical mixing theorem.

The new upper-bound source is a preprint. Please preserve the dataset's unrefereed-claim distinction. We checked its relevant argument and time convention, not an exhaustive novelty review. The original AIM page was unavailable; the question was compared against the frozen record and the v1.8 snapshot. This check is AI-assisted and self-audited, not independent human peer review.

Alper Ferudun · Mercury Software GmbH

OWR-13353-012 — Computing the Odd-Ear Prefix Parameter Is NP-Complete

I have published a five-page preprint addressing the computational-complexity question in this entry:
https://doi.org/10.5281/zenodo.23592264

The theorem states that deciding whether beta(G) >= k, with k part of the input, is NP-complete. Here beta counts the initial odd ears of an open ear decomposition, including its initial cycle. Hardness holds even for simple, 2-connected, 2-degenerate graphs that become bipartite after deleting one vertex, with girth at least any prescribed fixed bound. Exact computation is consequently NP-hard. This does not assert an unconditional separation of P and NP and does not conflict with the known fixed-parameter tractability result.

The argument identifies beta on a cone over a bipartite multigraph with a connected, perfectly matchable induced-edge maximization problem; a two-scale multiplicity reduction and an odd-subdivision invariance lemma give the hardness. The Benchetrit-Sebo/Lovasz theorem inputs and the connected-matching SAT gadget of Gomes and coauthors are explicitly credited. The original question is in Oberwolfach Report 51/2014, printed p. 2914, Question 4, and arXiv:1509.05586v1, Question 3.3.

The DOI record includes the PDF, portable LaTeX source, written audit, and executable exact diagnostics. The finite checks supplement the general proof; they are not formal verification. This is an AI-assisted, self-audited, unrefereed preprint, not an independently reviewed or priority-certified result. I report the claimed resolution of the complexity-classification question for consideration; no maintainer acceptance or dataset status change is implied.

Alper Ferudun — Mercury Software GmbH

Two scoped manuscript contributions, with no request to mark either source as newly solved:

AIM-COMBINATORICS-0025 — Exact Progression Lengths in Finite Interval-Digit Sets. For every finite initial-interval base-b digit alphabet, the manuscript gives an exact longest-progression formula and an explicit extremizer, together with the infinite limit, stabilization depth and iterated-sumset consequences. This is a prescribed-alphabet and finite-depth refinement of credited Walker–Walker methods. It contributes to the integer-sumset example-building part of the record; it does not solve the entire multi-part question or give a counterexample to a proved polynomial Freiman–Ruzsa theorem. The source remains partially_solved.

DOI and archived PDF/source/verification files: https://doi.org/10.5281/zenodo.23601470
Readable abstract and files: https://eulersolve.org/papers/aim-combinatorics-0025/

AIM-LINEAR_ALGEBRA-0010 — Exact Vanishing Orders for Exponentially Deformed Toeplitz Minors. For finite-atom Edrei Toeplitz matrices under exponential deformation, the manuscript gives an explicit formula for every admissible minor's vanishing order, a contiguous interval of strictly positive coefficients, and endpoint coefficients. The classical zero-pattern classification, supersymmetric Schur framework and qualitative strictification are credited prior work. This is an adjacent quantitative refinement, not a new solution of the source question, which remains solved in the literature.

DOI and archived PDF/source/verification files: https://doi.org/10.5281/zenodo.23601572
Readable abstract and files: https://eulersolve.org/papers/aim-linear-algebra-0010/

Both are English v1.0 preprints by Alper Ferudun, Mercury Software GmbH, released under CC BY 4.0. They are AI-assisted, self-audited and unrefereed. Publication does not establish independent peer review, formal verification, certified novelty, priority, or dataset-maintainer acceptance. Original-source closure increment: 0. The existing broader source classifications are preserved.

Scoped research update for AIM-COMBINATORICS-0014: Boundary Rigidity for Equi-isoclinic Planes.

For real two-dimensional subspaces with common isoclinic projection length alpha = 1/2, we prove that a negative-determinant triangle of normalized cross-Gram matrices forces the entire family into at most six dimensions, with at most nine planes. An explicit eight-type extension analysis classifies this reflection-triangle branch and gives one unordered congruence class at equality. The proof uses the anchor Gram null relation and exact polynomial compatibility identities; a reproducibility script over Q(sqrt(3)) is included.

Scope and attribution: the maximum of nine planes in R6 and its attainability were already known from Et-Taoui (2007); the complex-realification criterion and holonomy tools are credited to prior work. The claimed contribution is the ambient-independent reflection-branch rigidity and finite description. This does not resolve the general fixed-angle conjectures in the source record, the positive-triangle complex branch, or other parameters. The source should remain PARTIALLY_SOLVED, not be marked fully solved on the basis of this paper. The closest 2007 and 2024 full texts were unavailable to this bounded search, so precise novelty and absolute priority are not certified.

Preprint and reproducibility files: https://doi.org/10.5281/zenodo.23602906
Landing page: https://eulersolve.org/papers/aim-combinatorics-0014/

Author: Alper Ferudun, Mercury Software GmbH. English, version 1.0, CC BY 4.0. AI-assisted, self-audited and unrefereed; no independent review or formal verification is claimed. This is a scoped result report, not a request to change the whole source record to solved.

Scoped result for AIM-ALGEBRAIC_GEOMETRY-0185 (AIM Algebraic vision, Reconstruction, Problem 1.25; tracked against the frozen v1.6.0 entry): Base-Divisor Degree Formulas for Two-View Algebraic Curves, Alper Ferudun.

For two fixed projective cameras with distinct centers, and integral image curves that are not epipolar lines over an algebraically closed characteristic-zero field, the normalized components of the moving-pencil fiber product give all integral world reconstructions. If h is a component's pencil degree and B_X, B_Y are its pulled-back epipolar base divisors, its world degree is h + deg max(B_X, B_Y). The manuscript also proves a branch formula for the baseline cycle multiplicity. When both moving pencils have degree one, the reconstruction is unique and its possible degrees for fixed image degrees m,n are exactly max(m,n), ..., m+n-1.

This is a complete proof of the stated scoped algebraic theorems, not a claim that the whole open-ended source problem is solved. Real visibility, general smoothness, nonreduced inputs and the full baseline primary ideal are outside the result; our broader-source classification remains PARTIALLY_SOLVED. The classical fiber-product approach and twisted-cubic example are credited, including Kaminski--Fryers--Teicher (2005), Rydell--Sundelius (2024) and Ballico (2024). No absolute novelty or priority claim is made.

• DOI, PDF, source and exact-checking package: https://doi.org/10.5281/zenodo.23604002
• Paper page: https://eulersolve.org/papers/aim-algebraic-geometry-0185/
• Original source question: http://aimpl.org/algvision/1/

The six-page preprint is AI-assisted, self-audited and unrefereed. The 81 monomial-cone checks, six named examples, 285 sharpness constructions and 9,216 local-order checks are supporting evidence, not substitutes for the written proofs. Independent review, proof-assistant verification and maintainer acceptance are not claimed.